## Abstract

We establish existence, uniqueness, and Sobolev and Hölder regularity results for the stochastic partial differential equation du=(∑i,j=1da^{ij}u_{xixj}+f^{0}+∑i=1df_{xi}^{i})dt+∑k=1∞g^{k}dw_{t}^{k},t>0,x∈D given with non-zero initial data. Here {w_{t}^{k}:k=1,2,⋯} is a family of independent Wiener processes defined on a probability space (Ω,P), a^{ij}=a^{ij}(ω,t) are merely measurable functions on Ω×(0,∞), and D is either a polygonal domain in R^{2} or an arbitrary dimensional conic domain of the type [Formula presented] where M is an open subset of S^{d−1} with C^{2} boundary. We measure the Sobolev and Hölder regularities of arbitrary order derivatives of the solution using a system of mixed weights consisting of appropriate powers of the distance to the vertices and of the distance to the boundary. The ranges of admissible powers of the distance to the vertices and to the boundary are sharp.

Original language | English |
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Pages (from-to) | 463-520 |

Number of pages | 58 |

Journal | Journal of Differential Equations |

Volume | 340 |

DOIs | |

Publication status | Published - 2022 Dec 15 |

### Bibliographical note

Funding Information:The first and third authors were supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2020R1A2C1A01003354).The second author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2019R1F1A1058988).

Publisher Copyright:

© 2022 Elsevier Inc.

## Keywords

- Conic domains
- Mixed weight
- Parabolic equation
- Weighted Sobolev regularity

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics